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<div class="add">

<b>let </b><font color="Maroon" title="c1">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">q1</font>, <font color="Maroon" title="c5">q2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>and </b><br/><a NAME="E3:42"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>and </b><br/><a NAME="E4:42"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p1</font>
 <b>and </b><br/><a NAME="E5:42"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font>
 <b>and </b><br/><a NAME="E6:42"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p1</font>
 <b>and </b><br/><a NAME="E7:42"/><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font>
 <b>and </b><br/><a NAME="E8:42"/><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q2</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 </span><b>by </b><i><a class="txt" href="jordan16.html#E1:42"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="jordan16.html#E2:42"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="jordan16.html#E3:42"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="jordan16.html#E8:42"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="jordan5c.html#T14" title="JORDAN5C:th.14">JORDAN5C:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><b>suppose </b></a><a NAME="E1:42_1_1"/><i><font color="Green" title="E33">A9</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c6">S</font> =  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>;<br/><a NAME="E2:42_1_1"/>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>
 <b>by </b><i><a class="txt" href="jordan16.html#E1:42"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="jordan16.html#E8:42"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="jordan16.html#E1:42_1_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="jordan16.html#T36" target="_self" title="JORDAN16:th.36">Th36</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">S</font> =  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/><b>take </b>
<font color="Maroon" title="c7">S</font>
;<br/><b>thus </b><a NAME="E4:42_1_1"/>
<font color="Maroon" title="c7">S</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>
 <b>by </b><i><a class="txt" href="jordan16.html#E1:42"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="jordan16.html#E8:42"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="jordan16.html#E1:42_1_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="jordan16.html#T36" target="_self" title="JORDAN16:th.36">Th36</a></i>;<br/><b>thus </b><a NAME="E5:42_1_1"/>
<font color="Maroon" title="c7">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="jordan16.html#T5" target="_self" title="JORDAN16:th.5">Th5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_1"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c7">S</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/><a NAME="E2:42_1_1_1"/><i><font color="Green" title="E35">A11</font></i>: 
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> )
 </span><b>by </b><i><a class="txt" href="jordan16.html#E1:42_1_1_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T57" title="ZFMISC_1:th.57">ZFMISC_1:57</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 ;<br/><div class="add"><a NAME="E2:42_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 <b>by </b><i><a class="txt" href="jordan16.html#E2:42_1_1_1"><i><font color="Green" title="E35">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E3:42_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan16.html#E1:42"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="jordan16.html#E4:42"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="jordan6.html#T69" title="JORDAN6:th.69">JORDAN6:69</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_2"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 ;<br/><div class="add"><a NAME="E2:42_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 <b>by </b><i><a class="txt" href="jordan16.html#E2:42_1_1_1"><i><font color="Green" title="E35">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E3:42_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan16.html#E1:42"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="jordan16.html#E7:42"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="jordan6.html#T70" title="JORDAN6:th.70">JORDAN6:70</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:42_1_1"/>
<font color="Maroon" title="c7">S</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2"><b>suppose </b></a><a NAME="E1:42_1_2"/><i><font color="Green" title="E33">A12</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c6">S</font> =  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>;<br/><a NAME="E2:42_1_2"/>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>
 <b>by </b><i><a class="txt" href="jordan16.html#E1:42"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="jordan16.html#E8:42"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="jordan16.html#E1:42_1_2"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="jordan16.html#T36" target="_self" title="JORDAN16:th.36">Th36</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">S</font> =  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/><b>take </b>
<font color="Maroon" title="c7">S</font>
;<br/><a NAME="E4:42_1_2"/>
<font color="Maroon" title="c7">S</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c4">q1</font>
 <b>by </b><i><a class="txt" href="jordan16.html#E1:42"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="jordan16.html#E8:42"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="jordan16.html#E1:42_1_2"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="jordan16.html#T36" target="_self" title="JORDAN16:th.36">Th36</a></i>;<br/><b>hence </b><a NAME="E5:42_1_2"/>
<font color="Maroon" title="c7">S</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>
 <b>by </b><i><a class="ref" href="jordan5b.html#T14" title="JORDAN5B:th.14">JORDAN5B:14</a></i>;<br/><b>thus </b><a NAME="E6:42_1_2"/>
<font color="Maroon" title="c7">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="jordan16.html#T5" target="_self" title="JORDAN16:th.5">Th5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_2_1"/><i><font color="Green" title="E34">A13</font></i>: 
<font color="Maroon" title="c7">S</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/><a NAME="E2:42_1_2_1"/><i><font color="Green" title="E35">A14</font></i>: 
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_2_1_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> or <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> )
 </span><b>by </b><i><a class="txt" href="jordan16.html#E1:42_1_2_1"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#T57" title="ZFMISC_1:th.57">ZFMISC_1:57</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_2_1_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 ;<br/><div class="add"><a NAME="E2:42_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 <b>by </b><i><a class="txt" href="jordan16.html#E2:42_1_2_1"><i><font color="Green" title="E35">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E3:42_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan16.html#E1:42"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="jordan16.html#E6:42"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="jordan6.html#T69" title="JORDAN6:th.69">JORDAN6:69</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_2_1_1_2"/>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 ;<br/><div class="add"><a NAME="E2:42_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q2</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 <b>by </b><i><a class="txt" href="jordan16.html#E2:42_1_2_1"><i><font color="Green" title="E35">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E3:42_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan16.html#E1:42"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="jordan16.html#E5:42"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="jordan6.html#T70" title="JORDAN6:th.70">JORDAN6:70</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:42_1_2"/>
<font color="Maroon" title="c7">S</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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